Rank One Solvable p-adic Differential Equation and Finite Abelian Characters via Lubin-Tate groups

We introduce a new class of exponentials of Artin-Hasse type, called $\boldsymbol{\pi}$-exponentials. These exponentials depends on the choice of a generator $\boldsymbol{\pi}$ of the Tate module of a Lubin-Tate group $\mathfrak{G}$ over $\mathbb{Z}_p$. They arise naturally as solutions of solvable differential modules over the Robba ring. If $\mathfrak{G}$ is isomorphic to $\hat{\mathbb{G}}_m$ over $\mathbb{Z}_p$, we develop methods to test their over-convergence, and get in this way a stronger version of the Frobenius structure theorem for differential equations. We define a natural transformation of the Artin-Schreier complex into the Kummer complex. This provides an explicit generator of the Kummer unramified extension of $\mathcal{E}^{\dag}_{K_{\infty}}$, whose residue field is a given Artin-Schreier extension of k((t)), where k is the residue field of K. We then compute explicitely the group, under tensor product, of isomorphism classes of rank one solvable differential equations. Moreover, we get a canonical way to compute the rank one $\phi$-module over $\mathcal{E}^{\dag}_{K_{\infty}}$ attached to a rank one representation of $Gal(k((t))^{sep}/k((t)))$, defined by an Artin-Schreier character.